Then in AC her old jealous feelings are reawakened when she finds that Cloud is living at the church, for she gets the impression that he's drawn there for more reasons than just his guilt ("dragging the past around")
Dragging the past around is in reference to how guilty he feels about Aerith's death. I have no idea what makes you think otherwise.

When Cloud says in AC, that he doesn't think he can protect anyway, Tifa's response is "zuruzuru zuruzuru." which is an onomatopoeia for dragging.
ずるずる Nihongodict: Free Online English ⇆ Japanese Dictionary